$n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability

TL;DR

This paper studies n-dimensional position-dependent mass damped harmonic oscillators, demonstrating their linearizability via point transformations and providing exact solutions by mapping from constant mass cases.

Contribution

It introduces a method to linearize and solve n-dimensional PDM damped harmonic oscillators using point canonical transformations, extending known solutions to variable mass systems.

Findings

Linearizability of n-dimensional PDM DHOs achieved

Exact solutions mapped from constant mass to PDM systems

Phase-space trajectories illustrated for examples

Abstract

We consider position-dependent mass (PDM) Lagrangians/Hamiltonians in their standard textbook form, where the long-standing \emph{gain-loss balance} between the kinetic and potential energies is kept intact to allow conservation of total energy (i.e., $L=T-V$, $H=T+V$, and $dH/dt=dE/dt=0$). Under such standard settings, we discuss and report on $n$-dimensional PDM damped harmonic oscillators (DHO). We use some $n$-dimensional point canonical transformation to facilitate the linearizability of their $n$-PDM dynamical equations into some $n$-linear DHOs' dynamical equations for constant mass setting. Consequently, the well know exact solutions for the linear DHOs are mapped, with ease, onto the exact solutions for PDM DHOs. A set of one-dimensional and a set of $n$-dimensional PDM-DHO illustrative examples are reported along with their phase-space trajectories.